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This invention relates to prediction interval calculators and, more particularly, to a prediction interval calculator that performs a complete statistical analysis of the output of an inductively coupled plasma spectrophotometer in accordance with Six Sigma.
With the advent of the worldwide marketplace and the corresponding consumer demand for highly reliable products, quality has become an increasingly important issue. The quality of a company""s product line can therefore play a decisive role in determining the company""s reputation and profitability. As a result of this pressure for defect-free products, increased emphasis is being placed on quality control at all levels; it is no longer just an issue with which quality control managers are concerned. This has led to various initiatives designed to improve quality, such as the Total Quality Management (TQM) and the Six Sigma quality improvement programs. An overview of the Six Sigma program is presented by Mikel J. Harry and J. Ronald Lawson in xe2x80x9cSix Sigma Producibility Analysis and Process Characterization,xe2x80x9d Addison Wesley Publishing Co., pp.1-1 through 1-5, 1992. The Six Sigma program is also thoroughly discussed by G. J. Hahn, W. J. Hill, R. W. Hoerl, and S. A. Zinkgraf in xe2x80x9cThe Impact of Six Sigma Improvementxe2x80x94A Glimpse into the Future of Statisticsxe2x80x9d, The American Statistician, 53, 3, August, pages 208-215; and by G. J. Hahn, N. Doganaksoy, and R. Hoerl in xe2x80x9cThe Evolution of Six Sigmaxe2x80x9d, to appear in Quality Engineering, March 2000 issue.
Six Sigma analysis is a data driven methodology to improve the quality of products and services delivered to customers. Decisions made regarding direction, interpretation, scope, depth or any other aspect of quality effort should be based on actual data gathered, and not based on opinion, authority or guesswork. Key critical-to-quality (CTQ) characteristics are set by customers. Based on those CTQs, internal measurements and specifications are developed in order to quantify quality performance. Quality improvement programs are developed whenever there is a gap between the customer CTQ expectations and the current performance level.
The basic steps in a quality improvement project are first to define the real problem by identifying the CTQs and related measurable performance that is not meeting customer expectations. This real problem is then translated into a statistical problem through the collection of data related to the real problem. By the application of the scientific method (observation, hypothesis and experimentation), a statistical solution to this statistical problem is arrived at. This solution is deduced from the data through the testing of various hypotheses regarding a specific interpretation of the data. Confidence (prediction) intervals provide a key statistical tool used to accept or reject hypotheses that are to be tested. The arrived at statistical solution is then translated back to the customer in the form of a real solution.
In common use, data is interpreted on its face value. However, from a statistical point of view, the results of a measurement cannot be interpreted or compared without a consideration of the confidence that measurement accurately represents the underlying characteristic that is being measured. Uncertainties in measurements will arise from variability in sampling, the measurement method, operators and so forth. The statistical tool for expressing this uncertainty is called a confidence interval depending upon the exact situation in which the data is being generated.
Confidence interval refers to the region containing the limits or band of a parameter with an associated confidence level that the bounds are large enough to contain the true parameter value. The bands can be single-sided to describe an upper or lower limit or double sided to describe both upper and lower limits. The region gives a range of values, bounded below by a lower confidence limit and from above by an upper confidence limit, such that one can be confident (at a pre-specified level such as 95% or 99%) that the true population parameter value is included within the confidence interval. Confidence intervals can be formed for any of the parameters used to describe the characteristic of interest. In the end, confidence intervals are used to estimate the population parameters from the sample statistics and allow a probabilistic quantification of the strength of the best estimate.
In the case of the invention described herein, the calculated prediction intervals describe a range of values which contain the actual value of the sample at some given double-sided confidence level. For example, the present invention allows the user to change a statistically undependable statement, xe2x80x9cThere is 5.65 milligrams of Element Y in sample Xxe2x80x9d, to, xe2x80x9cThere is 95% confidence that there is 5.65+/xe2x88x920.63 milligrams of Element Y in sample Xxe2x80x9d. A prediction interval for an individual observation is an interval that will, with a specified degree of confidence, contain a randomly selected observation from a population. The inclusion of the confidence interval at a given probability allows the data to be interpreted in light of the situation. The interpreter has a range of values bounded by an upper and lower limit that is formed for any of the parameters used to describe the characteristic of interest. Meanwhile and at the same time, the risk associated with and reliability of the data is fully exposed allowing the interpreter access to all the information in the original measurement. This full disclosure of the data can then be used in subsequent decisions and interpretations of which the measurement data has bearing.
Current generation inductively coupled plasma spectrophotometers use a linear calibration scheme to back calculate sample concentrations based on known calibration samples. By spiking all samples with a constant level of a particular element (usually scandium or yttrium) this element can be used as an internal calibration by normalizing all other element readings of this element. After specifying acceptable tolerances for read back samples, a typical batch run may contain several calibrations performed during the course of the run. For example, calibration samples can be run a total of six times for a particular data set. Each sample concentration must then be back calculated for each calibration. However, the calculations routinely performed by these instruments do not calculate confidence intervals for the back calculations of each sample concentration.
To calculate these parameters can be cumbersome, even if a hand-held calculator is used. To avoid the inconvenience of using calculators, look-up tables are often used instead, in which the various parameters of interest are listed in column""s and correlated with each other. Nevertheless, these tables do not provide the user with enough flexibility, e.g., it is generally necessary to interpolate between the listed values. Furthermore, the user is not presented information in a way that is interactive, so that a xe2x80x9cfeelxe2x80x9dfor the numbers and the relationship of the various quantities to each other is lost.
Thus, there is a particular need for an apparatus and method for calculating confidence intervals under Six Sigma.
An exemplary embodiment of the present invention is an apparatus and method for calculating error analysis. A method for calculating at least one confidence interval comprises activating a calculator. The user selects an input data. The calculator reformats the input data to an output data. The calculator generates a specific calibration summary using said output data. The calculator plots at least one linear calibration curve derived from the specific calibration summary. The calculator then plots at least one residual calibration value plot derived from the specific calibration summary. The calculator calculates at least one back-calculated unknown sample value. The calculator then calculates at least one confidence interval for said at least one back-calculated unknown sample value.
An apparatus for calculating error analysis, the apparatus comprises a set of instructions for calculating at least one confidence interval value.
These and other features and advantages of the present invention will be apparent from the following brief description of the drawings, detailed description, and appended claims and drawings.